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Logic
Statements, negations, connectives, truth tables, equivalent statements, De Morgan’s Laws, arguments, Euler diagrams
Part 1: Statements, Negations, and Quantified Statements
A statement is a sentence that is either true or false but not both simultaneously.
Ex: a. Paris is the capital of France;
b. Edgar Poe wrote the last episode of Monk.
Commands, questions, and opinions, are not statements because they are neither true nor false.
Ex: a. Titanic is the greatest movie of all time. ( opinion)
b. Solve the exercises 20 – 50. (command) c. If I start losing my memory, how will I know? (question)
In symbolic logic, we use lowercase letters such as p, q, r , and s to represent statements.
Ex: p : Paris is the capital of France;
q : Edgar Poe wrote the last episode of Monk.
The negation of a statement has a meaning that is opposite that of the original meaning. The negation of a true statement is a false statement and the negation of a false statement is a true
statement
Ex: a. The negation of the statement “ Edgar Poe wrote the last episode of Monk ” can be
“ Edgar Poe did not write the last episode of Monk ” or also “ It is not true that Edgar Poe wrote
the last episode of Monk”
b. The negation of the statement “Today is not raining” is “ Today is raining ” or “ It is not
true that today is not raining”.
Symbolically, the negation of a statement p is denoted by ~p.
Ex: p : Today is Sunday;
~p : Today is not Sunday.
The words all , some , and no (or none) are called quantifiers.
Ex: Statements containing a quantifier:
All poets are writers. Some people are bigots. No math books have pictures. Some students do not work hard.
Equivalent Ways of Expressing Quantified Statements
Statement Equivalent way to express it Example All A are B There are no A that are no B All teachers are trained; There are no teachers that are not trained. Some A are B There is at least one A that is in B
Some people like ice-cream; At least one person likes ice- cream. No A are B All A are not B No car is running on the rail train; All the cars are not running on the rail train. Some A are not B Not all A are B Some students do not pass the class; Not all of the students are passing the class.
Part 2: Compound Statements and Connectives
Simple statements convey one idea with no connecting words.
Compound statements combine two or more simple statements using connectives. Connectives are words used to join simple statements. Examples are: and , or , if…then , and if and only if.
If p and q are two simple statements, then the compound statement “ p and q ” is symbolized
by p q****. The compound statement formed by connecting statements with the word and is called
a conjunction. The symbol for and is .
Ex: Let p and q represent the following simple statements:
p : It is Sunday.
q : They are working.
The compound statement “ It is Sunday and they are working ” can be formally/symbolically
expressed by “ p q ”.
The compound statement “ It is Sunday and they are not working ” can be formally/symbolically
expressed by “ p ~q ”.
Common English Expressions for p q
Symbolic Statement English Statement Example: p : It is Sunday. q : They are working.
p q p^ and^ q^ It is Sunday and they are
working.
p q p^ but^ q^ It is Sunday, but they are
working.
p q p^ yet^ q^ It is Sunday, yet they are
working.
p q p^ nevertheless^ q^ It is Sunday; nevertheless they
are working.
The connective or can mean two different things. Consider the statement “ I visited New York
City or Houston, TX .”
This statement can mean ( exclusive or ) “ I visited New York City or Houston, TX but not both.”
It can also mean ( inclusive or ) “ I visited New York City or Houston, TX or both.”
Disjunction is a compound statement formed using the inclusive or represented by the symbol
.
Thus, “ p or q or both ” is symbolized by p q.
Ex: Let p and q represent the following simple statements:
p : The student prepared for the Test.
q : The student passed the Test.
The statement “ The student prepared for the Test or the student passed the Test ” can be written
as “ p q ”.
The statement “ The student prepared for the Test or the student did not pass the Test ” can be
written as “ p ~q ”.
The compound statement “ If p , then q is symbolized by p q. This is called a conditional
statement. The statement before the is called the antecedent. The statement after the is
called the consequent.
Ex: This diagram shows a relationship that can be expressed 3 ways:
All men are mortal. There are no men that are not mortal. If a person is a men, then that person is mortal.
Mortals
Men
p q q^ is necessary and sufficient for p
Being the Independence Day is necessary and sufficient for being 4th^ of July.
Ex: Let p and q represent the following simple statements:
p : She is laughing.
q : She is happy.
~(p q): “ It is not true that she is laughing and is happy ”;
~p q: “S he is not laughing and she is happy ”;
~(p q): “ She is neither laughing nor happy ” or “” It is not true that she is laughing or is
happy ”
Expressing Symbolic Statements with Parentheses in English
Symbolic Statement Statements to Group Together English Translation
( q ~p ) ~r q ~p If^ q^ and not^ p , then not^ r.
q ( ~p ~r ) ~p ~r q , and if not^ p^ then not^ r.
Remark: When we translate the symbolic statement into English, the simple statements in
parentheses appear on the same side of the comma.
Ex: Let p, q, and r represent the following simple statements:
p : A student misses class.
q : A student studies.
r : A student fails.
a. ( q ~ p ) ~ r: “ If a student studies and does not miss class, then the student does not fail.”
b. q (~ p ~ r ): “ A student studies, and if the student does not miss class, then the student does not fail.”
If a symbolic statement appears without parentheses, statements before and after the most
dominant connective should be grouped.
The dominance of connectives used in symbolic logic is defined in the following order:
Most dominant: Biconditional ՞
Same level of dominance: Conjunction
Disjunction
Conditional ՜
Least dominant: Negation ~
Statement Most Dominant Connective Highlighted in Red
Statements Meaning Clarified with Grouping Symbols
Type of Statement
p q ~r p q ~r p (q ~r) Conditional
p q ~r p q ~r (p q) ~r Conditional
p q r p q r p (q r) Biconditional
p q r p q r (p q) r Biconditional
p ~q r s p ~q r s (p ~q) (r s) Conditional
p q r p q r The meaning is ambiguous.^?
Ex: Let p, q, and r represent the following simple statements.
p : I fail the course.
q : I study hard.
r : I pass the final.
a. “ I do not fail the course if and only if I study hard and I pass the final ” ~ p ( q r )
First list the simple statements on top and show all the possible truth values.
Second, make a column for p q and fill in the truth values.
Third, construct one more column for ~( p q ). The final column tells us that the statement is
false only when both p and q are true.
p q p q ~( p q )
T T T F
T F F T
F T F T
F F F T
A compound statement that is always true is called a tautology. From the table, that means that
on its column need to be only Ts, no Fs.
Ex: p : Brazosport College is a college. (true)
q : UHCL is an university. (true)
~( p q ): “ It is not true that BC is a college and UHCL is an university ”.
The compound statement ~( p q ) is not a tautology because on the last column it contains at
least one F.
Ex: A truth table for (~ p q ) ~ q :
p q ~ p (^) ~ p q ~ q (^) (~ p q ) ~ q
T T F T F F
T F F F T F
F T T T F F
F F T T T T
Ex: Construct a truth table for the following statement:
a. I study hard and ace the final, or I fail the course.
b. Suppose that you study hard, you do not ace the final and you fail the course. Under these
conditions, is this compound statement true or false?
First we represent our statements as follows:
p : I study hard.
q : I ace the final.
r : I fail the course.
Then we write the statement “ I study hard and ace the final, or I fail the course ” in symbolic
form: ሺ ሻݍ ݎ.
Third, we build the table with three entries ( p, q and r ):
p q r p q (p q) r
T T T T T
T T F T T
T F T F T
T F F F F
F T T F T
F T F F F
F F T F T
F F F F F
For part b, we have that p is True, q is false and r is true, which means we need to look on the
row #3. The conclusion is T, so that, under these conditions, the statement is True.
Not all the time we need to construct the truth table. We can determine the truth value of a
compound statement for a specific case in which the truth values of the simple statements are
known substituting the truth values of the simple statements into the symbolic form of the
compound statement and use the appropriate definitions to determine the truth value of the
compound statement.
Ex: On the previous example, part b, we set the truth values: p is True, q is false and r is true. So,
the statement ሺ ሻݍ ݎ has in fact the value ܶሺ ሻܨ ܶ ܨ ൌ ܶ ܶൌ.
p if and only if q: p q and q p
p q p q
T T T
T F F
F T F
F F T
The Biconditional is True only when the component statements have the same value.
Ex: You receive a letter that states that you have been assigned a Prize Entry Number –
- If your number matches the winning pre-selected number and you return the number
before the deadline, you will win $1,000,000.00.
Suppose that your number does not match the winning pre-selected number, you return the
number before the deadline and only win a free issue of a magazine. Under these conditions, can
you sue the credit card company for making a false claim?
Assign letters to the simple statements in the claim;
p : Your number matches the pre-selected number False
q : You return the number before the deadline True
r : You win the prize False
Write the underlined claim in the letter in symbolic form: (pq)՜r.
Substitute the truth values for p , q , and r to determine the truth value for the letter’s claim:
(FT)՜F, so we have F՜F which is True.
The truth-value analysis indicates that you cannot sue the credit card company for making a false
claim.
Part 5: Equivalent Statements and Variation of Conditional Statements
Equivalent compound statements are made up of the same simple statements and have the
same corresponding truth values for all true-false combinations of these simple statements.
If a compound statement is true, then its equivalent statement must also be true. If a compound statement is false, its equivalent statement must also be false.
Using the truth tables, the corresponding columns for the two statements must be identical.
The symbol which is used to show an equivalence is ؠ.
Ex: Show that p ~q and ~ p ~q are equivalent.
First we construct a truth table and see if the corresponding truth values are the same:
p q ~ q (^) p ~q ~ p (^) ~ p ~q
T T F T F T
T F T T F T
F T F F T F
F F T T T T
The two shaded columns are identical, so the statements are equivalent. We write this p ~q ؠ~ p
~q.
Ex: Show that p՜q ؠ ~q՜ ~p.
p q p^ ՜^ q^ ~ p ~q (^) ~ q ~p
T T T^ F F T
T F F F T F
F T T^ T F T
F F T^ T T T
The two shaded columns are identical, so the statements are equivalent
The statement ~q՜ ~p is called the contrapositive of the conditional p՜q. They are logical equivalent.
Ex: Let’s prove the equivalence: ~( p ՜ q) ؠ p ~q.
p q p ՜ q ~(p ՜ q) ~q p ~q.
T T T F F F
T F F T T T
F T T F F F
F F T F T F
The negation of a conditional statement can be expressed in the following way:
~( p ՜ q) ؠ p ~q.
Ex: The negation of the statement “ If too much homework is given, a class should not be taken”
can be formed using:
p: Too much homework is given,
q: A class should be taken.
The symbolic form is p ~ q. The negation of p ~ q is p ~(~ q ) which simplifies to p q.
Translating this one into English we have: “ Too much homework is given and a class should be
taken.”
De Morgan’s Laws:
1. ~(p q) ؠ ~p ~q;
2. ~(p q) ؠ ~p ~q;
Ex: The first law is shown in the truth table below.
Practice Ex: Prove the second De Morgan’s Law.
Ex: a. The statement is given: “ All students do homework on weekends and I do not ”
The negation is: “ Some students do not do homework on weekends or I do”
b. The statement is given: “ Some college professors are entertaining lecturers or I’m
bored .”
The negation is: “No college professors are entertaining lecturers and I’m not bored .”
Part 7: Arguments and Truth Tables
An Argument consists of two parts:
p q (^) p q ~(p q) ~p ~q (^) ~p ~q
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
Ex: Determine whether this argument is valid or invalid: “ There is no need for makeup because if
there is an absence then there is a need for makeup but there is no absence.”
p : There is an absence
q: There is a need for makeup.
Expressed symbolically:
If there is an absence then there is need for makeup p q
There is no absence.. ~ p
Therefore, there is no need for makeup. ~q
The argument is in the form of the fallacy of the inverse and is therefore, invalid.
Part 8: Arguments and Euler Diagrams
An Euler diagram is a technique for determining the validity of arguments whose premises
contain the words all, some, and no.
Euler Diagrams and Arguments:
- Make an Euler diagram for the first premise.
- Make an Euler diagram for the second premise on top of the one for the first premise.
- The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. If there is even one possible diagram that contradicts the conclusion, this indicates that the conclusion is not true in every case, so the argument is invalid.
Ex: Premise 1: All students who arrive late cannot take the test.
Premise 2: All students who cannot take the test are ineligible for a passing grade.
Conclusion: Therefore, all students who arrive late are ineligible for a passing grade.
Since there is only one possible diagram, and it illustrates the argument’s conclusion the
argument is valid.
Cannot take test Cannot take test
Ineligible for a passing grade
